How to calculate volume of an area bounded between axis and curve turned 360 degrees across the x axis using integration?

视频信息

答案文本 复制

视频字幕 复制

Welcome to calculating volumes of revolution using integration. When we have a curve y equals f of x and we rotate the area between this curve and the x-axis through 360 degrees around the x-axis, we create a three-dimensional solid. The disk method allows us to find the volume of this solid using integration. The disk method uses the formula V equals the integral from a to b of pi times f of x squared dx. Here, a and b are the x-coordinates that bound our region, f of x is our curve function, and pi times f of x squared represents the area of each circular disk. When we slice the solid perpendicular to the x-axis, each slice is a disk with radius equal to f of x. Let's walk through the four-step process for solving volume of revolution problems. Step one: identify the function y equals f of x that defines our curve. Step two: determine the limits of integration, the x-values a and b that bound our region. Step three: set up the integral using our disk method formula. Step four: evaluate the definite integral to find the numerical volume. This systematic approach ensures we don't miss any important details. Let's work through a complete example. We want to find the volume when y equals square root of x is rotated around the x-axis from x equals 0 to x equals 4. First, we set up our integral: V equals the integral from 0 to 4 of pi times square root of x squared dx. This simplifies to pi times the integral of x dx. Evaluating this gives us pi times x squared over 2, evaluated from 0 to 4. Substituting our limits, we get pi times 16 over 2 minus 0, which equals 8 pi cubic units. To summarize, the disk method for calculating volumes of revolution uses the integral formula V equals the integral from a to b of pi times f of x squared dx. Remember these key points: always square the function, include pi in your formula, use the correct integration limits, and check your units. The disk method works for any continuous function rotated around the x-axis, making it a powerful tool for solving three-dimensional volume problems using integration.